Changeset 34e0c32 in sasview
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- Mar 18, 2016 7:13:07 PM (9 years ago)
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- master, ESS_GUI, ESS_GUI_Docs, ESS_GUI_batch_fitting, ESS_GUI_bumps_abstraction, ESS_GUI_iss1116, ESS_GUI_iss879, ESS_GUI_iss959, ESS_GUI_opencl, ESS_GUI_ordering, ESS_GUI_sync_sascalc, costrafo411, magnetic_scatt, release-4.1.1, release-4.1.2, release-4.2.2, release_4.0.1, ticket-1009, ticket-1094-headless, ticket-1242-2d-resolution, ticket-1243, ticket-1249, ticket885, unittest-saveload
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src/sas/models/media/olddocs/model_functions.rst
r6f17afa r34e0c32 102 102 the particle 103 103 104 .. image:: img/olddocs/image001.PNG104 .. image:: ..\img\olddocs\image001.PNG 105 105 106 106 with 107 107 108 .. image:: img/olddocs/image002.PNG108 .. image:: ..\img\olddocs\image002.PNG 109 109 110 110 where |P0|\ *(q)* is the un-normalized form factor, |rho|\ *(r)* is the scattering length density at a given … … 114 114 by the particle volume fraction 115 115 116 .. image:: img/olddocs/image003.PNG116 .. image:: ..\img\olddocs\image003.PNG 117 117 118 118 Our so-called 1D scattering intensity functions provide *P(q)* for the case where the scatterer is randomly oriented. In … … 337 337 The 1D scattering intensity is calculated in the following way (Guinier, 1955) 338 338 339 .. image:: img/olddocs/image004.PNG339 .. image:: ..\img\olddocs\image004.PNG 340 340 341 341 where *scale* is a volume fraction, *V* is the volume of the scatterer, *r* is the radius of the sphere, *bkg* is … … 372 372 NIST (Kline, 2006). Figure 1 shows a comparison of the output of our model and the output of the NIST software. 373 373 374 .. image:: img/olddocs/image005.jpg374 .. image:: ..\img\olddocs\image005.jpg 375 375 376 376 Figure 1: Comparison of the DANSE scattering intensity for a sphere with the output of the NIST SANS analysis software. … … 391 391 solution 392 392 393 .. image:: img/olddocs/image006.PNG393 .. image:: ..\img\olddocs\image006.PNG 394 394 395 395 where *Sij* are the partial structure factors and *fi* are the scattering amplitudes of the particles. The subscript 1 … … 397 397 where *n* = the number density) is internally calculated based on 398 398 399 .. image:: img/olddocs/image007.PNG399 .. image:: ..\img\olddocs\image007.PNG 400 400 401 401 The 2D scattering intensity is the same as 1D, regardless of the orientation of the *q* vector which is defined as 402 402 403 .. image:: img/olddocs/image008.PNG403 .. image:: ..\img\olddocs\image008.PNG 404 404 405 405 The parameters of the BinaryHSModel are the following (in the names, *l* (or *ls*\ ) stands for larger spheres … … 419 419 ============== ======== ============= 420 420 421 .. image:: img/olddocs/image009.jpg421 .. image:: ..\img\olddocs\image009.jpg 422 422 423 423 *Figure. 1D plot using the default values above (w/200 data point).* … … 445 445 The scattering intensity *I(q)* is calculated as: 446 446 447 .. image:: img/olddocs/image010.PNG447 .. image:: ..\img\olddocs\image010.PNG 448 448 449 449 where the amplitude *A(q)* is given as the typical sphere scattering convoluted with a Gaussian to get a gradual 450 450 drop-off in the scattering length density 451 451 452 .. image:: img/olddocs/image011.PNG452 .. image:: ..\img\olddocs\image011.PNG 453 453 454 454 Here |A2|\ *(q)* is the form factor, *P(q)*. The scale is equivalent to the volume fraction of spheres, each of … … 471 471 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 472 472 473 .. image:: img/olddocs/image008.PNG473 .. image:: ..\img\olddocs\image008.PNG 474 474 475 475 This example dataset is produced by running the FuzzySphereModel, using 200 data points, *qmin* = 0.001 -1, … … 487 487 ============== ======== ============= 488 488 489 .. image:: img/olddocs/image012.jpg489 .. image:: ..\img\olddocs\image012.jpg 490 490 491 491 *Figure. 1D plot using the default values (w/200 data point).* … … 508 508 The structure is: 509 509 510 .. image:: img/olddocs/raspberry_pic.jpg510 .. image:: ..\img\olddocs\raspberry_pic.jpg 511 511 512 512 where *Ro* = the radius of the large sphere, *Rp* = the radius of the smaller sphere on the surface, |delta| = the … … 523 523 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 524 524 525 .. image:: img/olddocs/image008.PNG525 .. image:: ..\img\olddocs\image008.PNG 526 526 527 527 This example dataset is produced by running the RaspBerryModel, using 2000 data points, *qmin* = 0.0001 |Ang^-1|, … … 544 544 ============== ======== ============= 545 545 546 .. image:: img/olddocs/raspberry_plot.jpg546 .. image:: ..\img\olddocs\raspberry_plot.jpg 547 547 548 548 *Figure. 1D plot using the values of /2000 data points.* … … 568 568 The 1D scattering intensity is calculated in the following way (Guinier, 1955) 569 569 570 .. image:: img/olddocs/image013.PNG570 .. image:: ..\img\olddocs\image013.PNG 571 571 572 572 where *scale* is a scale factor, *Vs* is the volume of the outer shell, *Vc* is the volume of the core, *rs* is the … … 608 608 NIST (Kline, 2006). Figure 1 shows a comparison of the output of our model and the output of the NIST software. 609 609 610 .. image:: img/olddocs/image014.jpg610 .. image:: ..\img\olddocs\image014.jpg 611 611 612 612 Figure 1: Comparison of the SasView scattering intensity for a core-shell sphere with the output of the NIST SANS … … 669 669 *qmax* = 0.7 -1 and the above default values. 670 670 671 .. image:: img/olddocs/image015.jpg671 .. image:: ..\img\olddocs\image015.jpg 672 672 673 673 *Figure: 1D plot using the default values (w/200 data point).* … … 675 675 The scattering length density profile for the default sld values (w/ 4 shells). 676 676 677 .. image:: img/olddocs/image016.jpg677 .. image:: ..\img\olddocs\image016.jpg 678 678 679 679 *Figure: SLD profile against the radius of the sphere for default SLDs.* … … 704 704 The *I* :sub:`0` is calculated in the following way (King, 2002) 705 705 706 .. image:: img/olddocs/secondmeq1.jpg706 .. image:: ..\img\olddocs\secondmeq1.jpg 707 707 708 708 where *scale* is a scale factor, *poly* is the sld of the polymer (or surfactant) layer, *solv* is the sld of the … … 731 731 ============== ======== ============= 732 732 733 .. image:: img/olddocs/secongm_fig1.jpg733 .. image:: ..\img\olddocs\secongm_fig1.jpg 734 734 735 735 REFERENCE … … 747 747 solvent and the shells are interleaved with layers of solvent. For *N* = 1, this returns the VesicleModel (above). 748 748 749 .. image:: img/olddocs/image020.jpg749 .. image:: ..\img\olddocs\image020.jpg 750 750 751 751 The 2D scattering intensity is the same as 1D, regardless of the orientation of the *q* vector which is defined as 752 752 753 .. image:: img/olddocs/image008.PNG753 .. image:: ..\img\olddocs\image008.PNG 754 754 755 755 NB: The outer most radius (= *core_radius* + *n_pairs* \* *s_thickness* + (*n_pairs* - 1) \* *w_thickness*) is used … … 774 774 is the number of shells. 775 775 776 .. image:: img/olddocs/image021.jpg776 .. image:: ..\img\olddocs\image021.jpg 777 777 778 778 *Figure. 1D plot using the default values (w/200 data point).* … … 801 801 The 1D scattering intensity is calculated in the following way 802 802 803 .. image:: img/olddocs/image022.gif804 805 .. image:: img/olddocs/image023.gif803 .. image:: ..\img\olddocs\image022.gif 804 805 .. image:: ..\img\olddocs\image023.gif 806 806 807 807 where, for a spherically symmetric particle with a particle density |rho|\ *(r)* 808 808 809 .. image:: img/olddocs/image024.gif809 .. image:: ..\img\olddocs\image024.gif 810 810 811 811 so that 812 812 813 .. image:: img/olddocs/image025.gif814 815 .. image:: img/olddocs/image026.gif816 817 .. image:: img/olddocs/image027.gif813 .. image:: ..\img\olddocs\image025.gif 814 815 .. image:: ..\img\olddocs\image026.gif 816 817 .. image:: ..\img\olddocs\image027.gif 818 818 819 819 Here we assumed that the SLDs of the core and solvent are constant against *r*. … … 821 821 Now lets consider the SLD of a shell, *r*\ :sub:`shelli`, defined by 822 822 823 .. image:: img/olddocs/image028.gif823 .. image:: ..\img\olddocs\image028.gif 824 824 825 825 An example of a possible SLD profile is shown below where *sld_in_shelli* (|rho|\ :sub:`in`\ ) and … … 829 829 For \| *A* \| > 0, 830 830 831 .. image:: img/olddocs/image029.gif831 .. image:: ..\img\olddocs\image029.gif 832 832 833 833 For *A* ~ 0 (eg., *A* = -0.0001), this function converges to that of the linear SLD profile (ie, … … 835 835 so this case is equivalent to 836 836 837 .. image:: img/olddocs/image030.gif838 839 .. image:: img/olddocs/image031.gif840 841 .. image:: img/olddocs/image032.gif842 843 .. image:: img/olddocs/image033.gif837 .. image:: ..\img\olddocs\image030.gif 838 839 .. image:: ..\img\olddocs\image031.gif 840 841 .. image:: ..\img\olddocs\image032.gif 842 843 .. image:: ..\img\olddocs\image033.gif 844 844 845 845 For *A* = 0, the exponential function has no dependence on the radius (so that *sld_out_shell* (|rho|\ :sub:`out`) is … … 847 847 factor contributed by the shells is 848 848 849 .. image:: img/olddocs/image034.gif850 851 .. image:: img/olddocs/image035.gif849 .. image:: ..\img\olddocs\image034.gif 850 851 .. image:: ..\img\olddocs\image035.gif 852 852 853 853 In the equation 854 854 855 .. image:: img/olddocs/image036.gif855 .. image:: ..\img\olddocs\image036.gif 856 856 857 857 Finally, the form factor can be calculated by 858 858 859 .. image:: img/olddocs/image037.gif859 .. image:: ..\img\olddocs\image037.gif 860 860 861 861 where 862 862 863 .. image:: img/olddocs/image038.gif863 .. image:: ..\img\olddocs\image038.gif 864 864 865 865 and 866 866 867 .. image:: img/olddocs/image039.gif867 .. image:: ..\img\olddocs\image039.gif 868 868 869 869 The 2D scattering intensity is the same as *P(q)* above, regardless of the orientation of the *q* vector which is 870 870 defined as 871 871 872 .. image:: img/olddocs/image040.gif872 .. image:: ..\img\olddocs\image040.gif 873 873 874 874 NB: The outer most radius is used as the effective radius for *S(Q)* when *P(Q)* \* *S(Q)* is applied. … … 892 892 NB: *rad_core* represents the core radius (*R1*) and *thick_shell1* (*R2* - *R1*) is the thickness of the shell1, etc. 893 893 894 .. image:: img/olddocs/image041.jpg894 .. image:: ..\img\olddocs\image041.jpg 895 895 896 896 *Figure. 1D plot using the default values (w/400 point).* 897 897 898 .. image:: img/olddocs/image042.jpg898 .. image:: ..\img\olddocs\image042.jpg 899 899 900 900 *Figure. SLD profile from the default values.* … … 918 918 The 1D scattering intensity is calculated in the following way (Guinier, 1955) 919 919 920 .. image:: img/olddocs/image017.PNG920 .. image:: ..\img\olddocs\image017.PNG 921 921 922 922 where *scale* is a scale factor, *Vshell* is the volume of the shell, *V1* is the volume of the core, *V2* is the total … … 928 928 and a shell thickness, *t*. 929 929 930 .. image:: img/olddocs/image018.jpg930 .. image:: ..\img\olddocs\image018.jpg 931 931 932 932 The 2D scattering intensity is the same as *P(q)* above, regardless of the orientation of the *q* vector which is 933 933 defined as 934 934 935 .. image:: img/olddocs/image008.PNG935 .. image:: ..\img\olddocs\image008.PNG 936 936 937 937 NB: The outer most radius (= *radius* + *thickness*) is used as the effective radius for *S(Q)* when *P(Q)* \* *S(Q)* … … 953 953 NB: *radius* represents the core radius (*R1*) and the *thickness* (*R2* - *R1*) is the shell thickness. 954 954 955 .. image:: img/olddocs/image019.jpg955 .. image:: ..\img\olddocs\image019.jpg 956 956 957 957 *Figure. 1D plot using the default values (w/200 data point).* … … 984 984 The 1D scattering intensity is calculated in the following way: 985 985 986 .. image:: img/olddocs/image022.gif987 988 .. image:: img/olddocs/image043.gif986 .. image:: ..\img\olddocs\image022.gif 987 988 .. image:: ..\img\olddocs\image043.gif 989 989 990 990 where, for a spherically symmetric particle with a particle density |rho|\ *(r)* 991 991 992 .. image:: img/olddocs/image024.gif992 .. image:: ..\img\olddocs\image024.gif 993 993 994 994 so that 995 995 996 .. image:: img/olddocs/image044.gif997 998 .. image:: img/olddocs/image045.gif999 1000 .. image:: img/olddocs/image046.gif1001 1002 .. image:: img/olddocs/image047.gif1003 1004 .. image:: img/olddocs/image048.gif1005 1006 .. image:: img/olddocs/image027.gif996 .. image:: ..\img\olddocs\image044.gif 997 998 .. image:: ..\img\olddocs\image045.gif 999 1000 .. image:: ..\img\olddocs\image046.gif 1001 1002 .. image:: ..\img\olddocs\image047.gif 1003 1004 .. image:: ..\img\olddocs\image048.gif 1005 1006 .. image:: ..\img\olddocs\image027.gif 1007 1007 1008 1008 Here we assumed that the SLDs of the core and solvent are constant against *r*. The SLD at the interface between … … 1011 1011 1) Exp 1012 1012 1013 .. image:: img/olddocs/image049.gif1013 .. image:: ..\img\olddocs\image049.gif 1014 1014 1015 1015 2) Power-Law 1016 1016 1017 .. image:: img/olddocs/image050.gif1017 .. image:: ..\img\olddocs\image050.gif 1018 1018 1019 1019 3) Erf 1020 1020 1021 .. image:: img/olddocs/image051.gif1021 .. image:: ..\img\olddocs\image051.gif 1022 1022 1023 1023 The functions are normalized so that they vary between 0 and 1, and they are constrained such that the SLD is … … 1027 1027 to the form factor *P(q)* 1028 1028 1029 .. image:: img/olddocs/image052.gif1030 1031 .. image:: img/olddocs/image053.gif1032 1033 .. image:: img/olddocs/image054.gif1029 .. image:: ..\img\olddocs\image052.gif 1030 1031 .. image:: ..\img\olddocs\image053.gif 1032 1033 .. image:: ..\img\olddocs\image054.gif 1034 1034 1035 1035 where we assume that |rho|\ :sub:`inter_i`\ *(r)* can be approximately linear within a sub-layer *j*. … … 1037 1037 In the equation 1038 1038 1039 .. image:: img/olddocs/image055.gif1039 .. image:: ..\img\olddocs\image055.gif 1040 1040 1041 1041 Finally, the form factor can be calculated by 1042 1042 1043 .. image:: img/olddocs/image037.gif1043 .. image:: ..\img\olddocs\image037.gif 1044 1044 1045 1045 where 1046 1046 1047 .. image:: img/olddocs/image038.gif1047 .. image:: ..\img\olddocs\image038.gif 1048 1048 1049 1049 and 1050 1050 1051 .. image:: img/olddocs/image056.gif1051 .. image:: ..\img\olddocs\image056.gif 1052 1052 1053 1053 The 2D scattering intensity is the same as *P(q)* above, regardless of the orientation of the *q* vector which is 1054 1054 defined as 1055 1055 1056 .. image:: img/olddocs/image040.gif1056 .. image:: ..\img\olddocs\image040.gif 1057 1057 1058 1058 NB: The outer most radius is used as the effective radius for *S(Q)* when *P(Q)* \* *S(Q)* is applied. … … 1081 1081 NB: *rad_core0* represents the core radius (*R1*). 1082 1082 1083 .. image:: img/olddocs/image057.jpg1083 .. image:: ..\img\olddocs\image057.jpg 1084 1084 1085 1085 *Figure. 1D plot using the default values (w/400 point).* 1086 1086 1087 .. image:: img/olddocs/image058.jpg1087 .. image:: ..\img\olddocs\image058.jpg 1088 1088 1089 1089 *Figure. SLD profile from the default values.* … … 1104 1104 of each string is assumed to be negligible. 1105 1105 1106 .. image:: img/olddocs/linearpearls.jpg1106 .. image:: ..\img\olddocs\linearpearls.jpg 1107 1107 1108 1108 *2.1.12.1. Definition* … … 1110 1110 The output of the scattering intensity function for the LinearPearlsModel is given by (Dobrynin, 1996) 1111 1111 1112 .. image:: img/olddocs/linearpearl_eq1.gif1112 .. image:: ..\img\olddocs\linearpearl_eq1.gif 1113 1113 1114 1114 where the mass *m*\ :sub:`p` is (SLD\ :sub:`pearl` - SLD\ :sub:`solvent`) \* (volume of *N* pearls). V is the total … … 1133 1133 NB: *num_pearls* must be an integer. 1134 1134 1135 .. image:: img/olddocs/linearpearl_plot.jpg1135 .. image:: ..\img\olddocs\linearpearl_plot.jpg 1136 1136 1137 1137 REFERENCE … … 1150 1150 distance. 1151 1151 1152 .. image:: img/olddocs/pearl_fig.jpg1152 .. image:: ..\img\olddocs\pearl_fig.jpg 1153 1153 1154 1154 *2.1.13.1. Definition* … … 1156 1156 The output of the scattering intensity function for the PearlNecklaceModel is given by (Schweins, 2004) 1157 1157 1158 .. image:: img/olddocs/pearl_eq1.gif1158 .. image:: ..\img\olddocs\pearl_eq1.gif 1159 1159 1160 1160 where 1161 1161 1162 .. image:: img/olddocs/pearl_eq2.gif1163 1164 .. image:: img/olddocs/pearl_eq3.gif1165 1166 .. image:: img/olddocs/pearl_eq4.gif1167 1168 .. image:: img/olddocs/pearl_eq5.gif1169 1170 .. image:: img/olddocs/pearl_eq6.gif1162 .. image:: ..\img\olddocs\pearl_eq2.gif 1163 1164 .. image:: ..\img\olddocs\pearl_eq3.gif 1165 1166 .. image:: ..\img\olddocs\pearl_eq4.gif 1167 1168 .. image:: ..\img\olddocs\pearl_eq5.gif 1169 1170 .. image:: ..\img\olddocs\pearl_eq6.gif 1171 1171 1172 1172 and 1173 1173 1174 .. image:: img/olddocs/pearl_eq7.gif1174 .. image:: ..\img\olddocs\pearl_eq7.gif 1175 1175 1176 1176 where the mass *m*\ :sub:`i` is (SLD\ :sub:`i` - SLD\ :sub:`solvent`) \* (volume of the *N* pearls/rods). *V* is the … … 1198 1198 NB: *num_pearls* must be an integer. 1199 1199 1200 .. image:: img/olddocs/pearl_plot.jpg1200 .. image:: ..\img\olddocs\pearl_plot.jpg 1201 1201 1202 1202 REFERENCE … … 1219 1219 The output of the 2D scattering intensity function for oriented cylinders is given by (Guinier, 1955) 1220 1220 1221 .. image:: img/olddocs/image059.PNG1221 .. image:: ..\img\olddocs\image059.PNG 1222 1222 1223 1223 where 1224 1224 1225 .. image:: img/olddocs/image060.PNG1225 .. image:: ..\img\olddocs\image060.PNG 1226 1226 1227 1227 and |alpha| is the angle between the axis of the cylinder and the *q*-vector, *V* is the volume of the cylinder, … … 1232 1232 and |phi|. Those angles are defined in Figure 1. 1233 1233 1234 .. image:: img/olddocs/image061.jpg1234 .. image:: ..\img\olddocs\image061.jpg 1235 1235 1236 1236 *Figure 1. Definition of the angles for oriented cylinders.* 1237 1237 1238 .. image:: img/olddocs/image062.jpg1238 .. image:: ..\img\olddocs\image062.jpg 1239 1239 1240 1240 *Figure 2. Examples of the angles for oriented pp against the detector plane.* … … 1259 1259 The output of the 1D scattering intensity function for randomly oriented cylinders is then given by 1260 1260 1261 .. image:: img/olddocs/image063.PNG1261 .. image:: ..\img\olddocs\image063.PNG 1262 1262 1263 1263 The *cyl_theta* and *cyl_phi* parameter are not used for the 1D output. Our implementation of the scattering kernel … … 1269 1269 NIST (Kline, 2006). Figure 3 shows a comparison of the 1D output of our model and the output of the NIST software. 1270 1270 1271 .. image:: img/olddocs/image065.jpg1271 .. image:: ..\img\olddocs\image065.jpg 1272 1272 1273 1273 *Figure 3: Comparison of the SasView scattering intensity for a cylinder with the output of the NIST SANS analysis* … … 1277 1277 In general, averaging over a distribution of orientations is done by evaluating the following 1278 1278 1279 .. image:: img/olddocs/image064.PNG1279 .. image:: ..\img\olddocs\image064.PNG 1280 1280 1281 1281 where *p(*\ |theta|,\ |phi|\ *)* is the probability distribution for the orientation and |P0|\ *(q,*\ |alpha|\ *)* is … … 1284 1284 distribution *p(*\ |theta|,\ |phi|\ *)* = 1.0. Figure 4 shows the result of such a cross-check. 1285 1285 1286 .. image:: img/olddocs/image066.jpg1286 .. image:: ..\img\olddocs\image066.jpg 1287 1287 1288 1288 *Figure 4: Comparison of the intensity for uniformly distributed cylinders calculated from our 2D model and the* … … 1309 1309 The 1D scattering intensity is calculated in the following way (Guinier, 1955) 1310 1310 1311 .. image:: img/olddocs/image072.PNG1311 .. image:: ..\img\olddocs\image072.PNG 1312 1312 1313 1313 where *scale* is a scale factor, *J1* is the 1st order Bessel function, *J1(x)* = (sin *x* - *x* cos *x*)/ *x*\ :sup:`2`. … … 1334 1334 ============== ======== ============= 1335 1335 1336 .. image:: img/olddocs/image074.jpg1336 .. image:: ..\img\olddocs\image074.jpg 1337 1337 1338 1338 *Figure. 1D plot using the default values (w/1000 data point).* … … 1341 1341 (Kline, 2006). 1342 1342 1343 .. image:: img/olddocs/image061.jpg1343 .. image:: ..\img\olddocs\image061.jpg 1344 1344 1345 1345 *Figure. Definition of the angles for the oriented HollowCylinderModel.* 1346 1346 1347 .. image:: img/olddocs/image062.jpg1347 .. image:: ..\img\olddocs\image062.jpg 1348 1348 1349 1349 *Figure. Examples of the angles for oriented pp against the detector plane.* … … 1371 1371 The Capped Cylinder geometry is defined as 1372 1372 1373 .. image:: img/olddocs/image112.jpg1373 .. image:: ..\img\olddocs\image112.jpg 1374 1374 1375 1375 where *r* is the radius of the cylinder. All other parameters are as defined in the diagram. Since the end cap radius … … 1380 1380 The scattered intensity *I(q)* is calculated as 1381 1381 1382 .. image:: img/olddocs/image113.jpg1382 .. image:: ..\img\olddocs\image113.jpg 1383 1383 1384 1384 where the amplitude *A(q)* is given as 1385 1385 1386 .. image:: img/olddocs/image114.jpg1386 .. image:: ..\img\olddocs\image114.jpg 1387 1387 1388 1388 The < > brackets denote an average of the structure over all orientations. <\ *A*\ :sup:`2`\ *(q)*> is then the form … … 1392 1392 The volume of the Capped Cylinder is (with *h* as a positive value here) 1393 1393 1394 .. image:: img/olddocs/image115.jpg1394 .. image:: ..\img\olddocs\image115.jpg 1395 1395 1396 1396 and its radius-of-gyration 1397 1397 1398 .. image:: img/olddocs/image116.jpg1398 .. image:: ..\img\olddocs\image116.jpg 1399 1399 1400 1400 **The requirement that** *R* >= *r* **is not enforced in the model! It is up to you to restrict this during analysis.** … … 1415 1415 ============== ======== ============= 1416 1416 1417 .. image:: img/olddocs/image117.jpg1417 .. image:: ..\img\olddocs\image117.jpg 1418 1418 1419 1419 *Figure. 1D plot using the default values (w/256 data point).* … … 1422 1422 |theta| = 45 deg and |phi| =0 deg with default values for other parameters 1423 1423 1424 .. image:: img/olddocs/image118.jpg1424 .. image:: ..\img\olddocs\image118.jpg 1425 1425 1426 1426 *Figure. 2D plot (w/(256X265) data points).* 1427 1427 1428 .. image:: img/olddocs/image061.jpg1428 .. image:: ..\img\olddocs\image061.jpg 1429 1429 1430 1430 *Figure. Definition of the angles for oriented 2D cylinders.* 1431 1431 1432 .. image:: img/olddocs/image062.jpg1432 .. image:: ..\img\olddocs\image062.jpg 1433 1433 1434 1434 *Figure. Examples of the angles for oriented pp against the detector plane.* … … 1453 1453 The output of the 2D scattering intensity function for oriented core-shell cylinders is given by (Kline, 2006) 1454 1454 1455 .. image:: img/olddocs/image067.PNG1455 .. image:: ..\img\olddocs\image067.PNG 1456 1456 1457 1457 where 1458 1458 1459 .. image:: img/olddocs/image068.PNG1460 1461 .. image:: img/olddocs/image239.PNG1459 .. image:: ..\img\olddocs\image068.PNG 1460 1461 .. image:: ..\img\olddocs\image239.PNG 1462 1462 1463 1463 and |alpha| is the angle between the axis of the cylinder and the *q*\ -vector, *Vs* is the volume of the outer shell … … 1468 1468 the outer shell is given by *L+2t*. *J1* is the first order Bessel function. 1469 1469 1470 .. image:: img/olddocs/image069.jpg1470 .. image:: ..\img\olddocs\image069.jpg 1471 1471 1472 1472 To provide easy access to the orientation of the core-shell cylinder, we define the axis of the cylinder using two … … 1503 1503 NIST (Kline, 2006). Figure 1 shows a comparison of the 1D output of our model and the output of the NIST software. 1504 1504 1505 .. image:: img/olddocs/image070.jpg1505 .. image:: ..\img\olddocs\image070.jpg 1506 1506 1507 1507 *Figure 1: Comparison of the SasView scattering intensity for a core-shell cylinder with the output of the NIST SANS* … … 1514 1514 2D output using a uniform distribution *p(*\ |theta|,\ |phi|\ *)* = 1.0. Figure 2 shows the result of such a cross-check. 1515 1515 1516 .. image:: img/olddocs/image071.jpg1516 .. image:: ..\img\olddocs\image071.jpg 1517 1517 1518 1518 *Figure 2: Comparison of the intensity for uniformly distributed core-shell cylinders calculated from our 2D model and* … … 1521 1521 *Solvent_sld* = 1e-6 |Ang^-2|, and *Background* = 0.0 |cm^-1|. 1522 1522 1523 .. image:: img/olddocs/image061.jpg1523 .. image:: ..\img\olddocs\image061.jpg 1524 1524 1525 1525 *Figure. Definition of the angles for oriented core-shell cylinders.* 1526 1526 1527 .. image:: img/olddocs/image062.jpg1527 .. image:: ..\img\olddocs\image062.jpg 1528 1528 1529 1529 *Figure. Examples of the angles for oriented pp against the detector plane.* … … 1545 1545 to any of the orientation angles, and also for the minor radius and the ratio of the ellipse radii. 1546 1546 1547 .. image:: img/olddocs/image098.gif1547 .. image:: ..\img\olddocs\image098.gif 1548 1548 1549 1549 *Figure.* *a* = *r_minor* and |nu|\ :sub:`n` = *r_ratio* (i.e., *r_major* / *r_minor*). … … 1551 1551 The function calculated is 1552 1552 1553 .. image:: img/olddocs/image099.PNG1553 .. image:: ..\img\olddocs\image099.PNG 1554 1554 1555 1555 with the functions 1556 1556 1557 .. image:: img/olddocs/image100.PNG1557 .. image:: ..\img\olddocs\image100.PNG 1558 1558 1559 1559 and the angle |bigpsi| is defined as the orientation of the major axis of the ellipse with respect to the vector *q*\ . … … 1574 1574 All angle parameters are valid and given only for 2D calculation; ie, an oriented system. 1575 1575 1576 .. image:: img/olddocs/image101.jpg1576 .. image:: ..\img\olddocs\image101.jpg 1577 1577 1578 1578 *Figure. Definition of angles for 2D* 1579 1579 1580 .. image:: img/olddocs/image062.jpg1580 .. image:: ..\img\olddocs\image062.jpg 1581 1581 1582 1582 *Figure. Examples of the angles for oriented elliptical cylinders against the detector plane.* … … 1597 1597 ============== ======== ============= 1598 1598 1599 .. image:: img/olddocs/image102.jpg1599 .. image:: ..\img\olddocs\image102.jpg 1600 1600 1601 1601 *Figure. 1D plot using the default values (w/1000 data point).* … … 1608 1608 and 76 degrees are taken for the angles of |theta|, |phi|, and |bigpsi| respectively). 1609 1609 1610 .. image:: img/olddocs/image103.gif1610 .. image:: ..\img\olddocs\image103.gif 1611 1611 1612 1612 *Figure. Comparison between 1D and averaged 2D.* … … 1615 1615 the results of the averaging by varying the number of angular bins. 1616 1616 1617 .. image:: img/olddocs/image104.gif1617 .. image:: ..\img\olddocs\image104.gif 1618 1618 1619 1619 *Figure. The intensities averaged from 2D over different numbers of bins and angles.* … … 1639 1639 The 2D scattering intensity is the same as 1D, regardless of the orientation of the *q* vector which is defined as 1640 1640 1641 .. image:: img/olddocs/image040.gif1641 .. image:: ..\img\olddocs\image040.gif 1642 1642 1643 1643 *2.1.19.1. Definition* 1644 1644 1645 .. image:: img/olddocs/image075.jpg1645 .. image:: ..\img\olddocs\image075.jpg 1646 1646 1647 1647 The chain of contour length, *L*, (the total length) can be described as a chain of some number of locally stiff … … 1666 1666 ============== ======== ============= 1667 1667 1668 .. image:: img/olddocs/image076.jpg1668 .. image:: ..\img\olddocs\image076.jpg 1669 1669 1670 1670 *Figure. 1D plot using the default values (w/1000 data point).* … … 1721 1721 - The scattering function is negative for a range of parameter values and q-values that are experimentally accessible. A correction function has been added to give the proper behavior. 1722 1722 1723 .. image:: img/olddocs/image077.jpg1723 .. image:: ..\img\olddocs\image077.jpg 1724 1724 1725 1725 The chain of contour length, *L*, (the total length) can be described as a chain of some number of locally stiff … … 1745 1745 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 1746 1746 1747 .. image:: img/olddocs/image008.PNG1747 .. image:: ..\img\olddocs\image008.PNG 1748 1748 1749 1749 This example dataset is produced by running the Macro FlexCylEllipXModel, using 200 data points, *qmin* = 0.001 |Ang^-1|, … … 1763 1763 ============== ======== ============= 1764 1764 1765 .. image:: img/olddocs/image078.jpg1765 .. image:: ..\img\olddocs\image078.jpg 1766 1766 1767 1767 *Figure. 1D plot using the default values (w/200 data points).* … … 1790 1790 and SLDs. 1791 1791 1792 .. image:: img/olddocs/image240.png1792 .. image:: ..\img\olddocs\image240.png 1793 1793 1794 1794 *(Graphic from DOI: 10.1039/C0NP00002G)* … … 1818 1818 and the 1D scattering intensity use the c-library from NIST. 1819 1819 1820 .. image:: img/olddocs/cscylbicelle_pic.jpg1820 .. image:: ..\img\olddocs\cscylbicelle_pic.jpg 1821 1821 1822 1822 *Figure. 1D plot using the default values (w/200 data point).* 1823 1823 1824 .. image:: img/olddocs/image061.jpg1824 .. image:: ..\img\olddocs\image061.jpg 1825 1825 1826 1826 *Figure. Definition of the angles for the oriented CoreShellBicelleModel.* 1827 1827 1828 .. image:: img/olddocs/image062.jpg1828 .. image:: ..\img\olddocs\image062.jpg 1829 1829 1830 1830 *Figure. Examples of the angles for oriented pp against the detector plane.* … … 1852 1852 The barbell geometry is defined as 1853 1853 1854 .. image:: img/olddocs/image105.jpg1854 .. image:: ..\img\olddocs\image105.jpg 1855 1855 1856 1856 where *r* is the radius of the cylinder. All other parameters are as defined in the diagram. … … 1863 1863 The scattered intensity *I(q)* is calculated as 1864 1864 1865 .. image:: img/olddocs/image106.PNG1865 .. image:: ..\img\olddocs\image106.PNG 1866 1866 1867 1867 where the amplitude *A(q)* is given as 1868 1868 1869 .. image:: img/olddocs/image107.PNG1869 .. image:: ..\img\olddocs\image107.PNG 1870 1870 1871 1871 The < > brackets denote an average of the structure over all orientations. <*A* :sup:`2`\ *(q)*> is then the form … … 1875 1875 The volume of the barbell is 1876 1876 1877 .. image:: img/olddocs/image108.jpg1877 .. image:: ..\img\olddocs\image108.jpg 1878 1878 1879 1879 1880 1880 and its radius-of-gyration is 1881 1881 1882 .. image:: img/olddocs/image109.jpg1882 .. image:: ..\img\olddocs\image109.jpg 1883 1883 1884 1884 **The requirement that** *R* >= *r* **is not enforced in the model!** It is up to you to restrict this during analysis. … … 1899 1899 ============== ======== ============= 1900 1900 1901 .. image:: img/olddocs/image110.jpg1901 .. image:: ..\img\olddocs\image110.jpg 1902 1902 1903 1903 *Figure. 1D plot using the default values (w/256 data point).* … … 1906 1906 |theta| = 45 deg and |phi| = 0 deg with default values for other parameters 1907 1907 1908 .. image:: img/olddocs/image111.jpg1908 .. image:: ..\img\olddocs\image111.jpg 1909 1909 1910 1910 *Figure. 2D plot (w/(256X265) data points).* 1911 1911 1912 .. image:: img/olddocs/image061.jpg1912 .. image:: ..\img\olddocs\image061.jpg 1913 1913 1914 1914 *Figure. Examples of the angles for oriented pp against the detector plane.* 1915 1915 1916 .. image:: img/olddocs/image062.jpg1916 .. image:: ..\img\olddocs\image062.jpg 1917 1917 1918 1918 Figure. Definition of the angles for oriented 2D barbells. … … 1940 1940 The 2D scattering intensity is the same as 1D, regardless of the orientation of the *q* vector which is defined as 1941 1941 1942 .. image:: img/olddocs/image008.PNG1942 .. image:: ..\img\olddocs\image008.PNG 1943 1943 1944 1944 The returned value is in units of |cm^-1| |sr^-1|, on absolute scale. … … 1946 1946 *2.1.23.1 Definition* 1947 1947 1948 .. image:: img/olddocs/image079.gif1948 .. image:: ..\img\olddocs\image079.gif 1949 1949 1950 1950 The scattering intensity *I(q)* is 1951 1951 1952 .. image:: img/olddocs/image081.PNG1952 .. image:: ..\img\olddocs\image081.PNG 1953 1953 1954 1954 where the contrast 1955 1955 1956 .. image:: img/olddocs/image082.PNG1956 .. image:: ..\img\olddocs\image082.PNG 1957 1957 1958 1958 and *N* is the number of discs per unit volume, |alpha| is the angle between the axis of the disc and *q*, and *Vt* 1959 1959 and *Vc* are the total volume and the core volume of a single disc, respectively. 1960 1960 1961 .. image:: img/olddocs/image083.PNG1961 .. image:: ..\img\olddocs\image083.PNG 1962 1962 1963 1963 where *d* = thickness of the layer (*layer_thick*), 2\ *h* = core thickness (*core_thick*), and *R* = radius of the 1964 1964 disc (*radius*). 1965 1965 1966 .. image:: img/olddocs/image084.PNG1966 .. image:: ..\img\olddocs\image084.PNG 1967 1967 1968 1968 where *n* = the total number of the disc stacked (*n_stacking*), *D* = the next neighbor center-to-center distance … … 1990 1990 ============== ======== ============= 1991 1991 1992 .. image:: img/olddocs/image085.jpg1992 .. image:: ..\img\olddocs\image085.jpg 1993 1993 1994 1994 *Figure. 1D plot using the default values (w/1000 data point).* 1995 1995 1996 .. image:: img/olddocs/image086.jpg1996 .. image:: ..\img\olddocs\image086.jpg 1997 1997 1998 1998 *Figure. Examples of the angles for oriented stackeddisks against the detector plane.* 1999 1999 2000 .. image:: img/olddocs/image062.jpg2000 .. image:: ..\img\olddocs\image062.jpg 2001 2001 2002 2002 *Figure. Examples of the angles for oriented pp against the detector plane.* … … 2021 2021 This model provides the form factor, *P(q)*, for a 'pringle' or 'saddle-shaped' object (a hyperbolic paraboloid). 2022 2022 2023 .. image:: img/olddocs/image241.png2023 .. image:: ..\img\olddocs\image241.png 2024 2024 2025 2025 *(Graphic from Matt Henderson, matt@matthen.com)* … … 2029 2029 The form factor calculated is 2030 2030 2031 .. image:: img/olddocs/pringle_eqn_1.jpg2031 .. image:: ..\img\olddocs\pringle_eqn_1.jpg 2032 2032 2033 2033 where 2034 2034 2035 .. image:: img/olddocs/pringle_eqn_2.jpg2035 .. image:: ..\img\olddocs\pringle_eqn_2.jpg 2036 2036 2037 2037 The parameters of the model and a plot comparing the pringle model with the equivalent cylinder are shown below. … … 2050 2050 ============== ======== ============= 2051 2051 2052 .. image:: img/olddocs/pringle-vs-cylinder.png2052 .. image:: ..\img\olddocs\pringle-vs-cylinder.png 2053 2053 2054 2054 *Figure. 1D plot using the default values (w/150 data point).* … … 2071 2071 The output of the 2D scattering intensity function for oriented ellipsoids is given by (Feigin, 1987) 2072 2072 2073 .. image:: img/olddocs/image059.PNG2073 .. image:: ..\img\olddocs\image059.PNG 2074 2074 2075 2075 where 2076 2076 2077 .. image:: img/olddocs/image119.PNG2077 .. image:: ..\img\olddocs\image119.PNG 2078 2078 2079 2079 and 2080 2080 2081 .. image:: img/olddocs/image120.PNG2081 .. image:: ..\img\olddocs\image120.PNG 2082 2082 2083 2083 |alpha| is the angle between the axis of the ellipsoid and the *q*\ -vector, *V* is the volume of the ellipsoid, *Ra* … … 2111 2111 above. 2112 2112 2113 .. image:: img/olddocs/image121.jpg2113 .. image:: ..\img\olddocs\image121.jpg 2114 2114 2115 2115 The *axis_theta* and *axis_phi* parameters are not used for the 1D output. Our implementation of the scattering 2116 2116 kernel and the 1D scattering intensity use the c-library from NIST. 2117 2117 2118 .. image:: img/olddocs/image122.jpg2118 .. image:: ..\img\olddocs\image122.jpg 2119 2119 2120 2120 *Figure. The angles for oriented ellipsoid.* … … 2126 2126 software. 2127 2127 2128 .. image:: img/olddocs/image123.jpg2128 .. image:: ..\img\olddocs\image123.jpg 2129 2129 2130 2130 *Figure 1: Comparison of the SasView scattering intensity for an ellipsoid with the output of the NIST SANS analysis* … … 2137 2137 cross-check. 2138 2138 2139 .. image:: img/olddocs/image124.jpg2139 .. image:: ..\img\olddocs\image124.jpg 2140 2140 2141 2141 *Figure 2: Comparison of the intensity for uniformly distributed ellipsoids calculated from our 2D model and the* … … 2169 2169 all orientations for 1D. 2170 2170 2171 .. image:: img/olddocs/image125.gif2171 .. image:: ..\img\olddocs\image125.gif 2172 2172 2173 2173 The returned value is in units of |cm^-1|, on absolute scale. … … 2177 2177 The form factor calculated is 2178 2178 2179 .. image:: img/olddocs/image126.PNG2179 .. image:: ..\img\olddocs\image126.PNG 2180 2180 2181 2181 To provide easy access to the orientation of the core-shell ellipsoid, we define the axis of the solid ellipsoid using … … 2203 2203 ============== ======== ============= 2204 2204 2205 .. image:: img/olddocs/image127.jpg2205 .. image:: ..\img\olddocs\image127.jpg 2206 2206 2207 2207 *Figure. 1D plot using the default values (w/1000 data point).* 2208 2208 2209 .. image:: img/olddocs/image122.jpg2209 .. image:: ..\img\olddocs\image122.jpg 2210 2210 2211 2211 *Figure. The angles for oriented CoreShellEllipsoid.* … … 2234 2234 *2.1.27.1. Definition* 2235 2235 2236 .. image:: img/olddocs/image125.gif2236 .. image:: ..\img\olddocs\image125.gif 2237 2237 2238 2238 The geometric parameters of this model are … … 2301 2301 where the volume *V* = (4/3)\ |pi| (*Ra* *Rb* *Rc*), and the averaging < > is applied over all orientations for 1D. 2302 2302 2303 .. image:: img/olddocs/image128.jpg2303 .. image:: ..\img\olddocs\image128.jpg 2304 2304 2305 2305 The returned value is in units of |cm^-1|, on absolute scale. … … 2309 2309 The form factor calculated is 2310 2310 2311 .. image:: img/olddocs/image129.PNG2311 .. image:: ..\img\olddocs\image129.PNG 2312 2312 2313 2313 To provide easy access to the orientation of the triaxial ellipsoid, we define the axis of the cylinder using the … … 2337 2337 ============== ======== ============= 2338 2338 2339 .. image:: img/olddocs/image130.jpg2339 .. image:: ..\img\olddocs\image130.jpg 2340 2340 2341 2341 *Figure. 1D plot using the default values (w/1000 data point).* … … 2348 2348 angles of |theta|, |phi|, and |psi| respectively). 2349 2349 2350 .. image:: img/olddocs/image131.gif2350 .. image:: ..\img\olddocs\image131.gif 2351 2351 2352 2352 *Figure. Comparison between 1D and averaged 2D.* 2353 2353 2354 .. image:: img/olddocs/image132.jpg2354 .. image:: ..\img\olddocs\image132.jpg 2355 2355 2356 2356 *Figure. The angles for oriented ellipsoid.* … … 2377 2377 The scattering intensity *I(q)* is 2378 2378 2379 .. image:: img/olddocs/image133.PNG2379 .. image:: ..\img\olddocs\image133.PNG 2380 2380 2381 2381 The form factor is 2382 2382 2383 .. image:: img/olddocs/image134.PNG2383 .. image:: ..\img\olddocs\image134.PNG 2384 2384 2385 2385 where |delta| = bilayer thickness. … … 2387 2387 The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 2388 2388 2389 .. image:: img/olddocs/image040.gif2389 .. image:: ..\img\olddocs\image040.gif 2390 2390 2391 2391 The returned value is in units of |cm^-1|, on absolute scale. In the parameters, *sld_bi* = SLD of the bilayer, … … 2402 2402 ============== ======== ============= 2403 2403 2404 .. image:: img/olddocs/image135.jpg2404 .. image:: ..\img\olddocs\image135.jpg 2405 2405 2406 2406 *Figure. 1D plot using the default values (w/1000 data point).* … … 2428 2428 The scattering intensity *I(q)* is 2429 2429 2430 .. image:: img/olddocs/image136.PNG2430 .. image:: ..\img\olddocs\image136.PNG 2431 2431 2432 2432 The form factor is 2433 2433 2434 .. image:: img/olddocs/image137.jpg2434 .. image:: ..\img\olddocs\image137.jpg 2435 2435 2436 2436 where |delta|\ T = tail length (or *t_length*), |delta|\ H = head thickness (or *h_thickness*), … … 2439 2439 The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 2440 2440 2441 .. image:: img/olddocs/image040.gif2441 .. image:: ..\img\olddocs\image040.gif 2442 2442 2443 2443 The returned value is in units of |cm^-1|, on absolute scale. In the parameters, *sld_tail* = SLD of the tail group, … … 2456 2456 ============== ======== ============= 2457 2457 2458 .. image:: img/olddocs/image138.jpg2458 .. image:: ..\img\olddocs\image138.jpg 2459 2459 2460 2460 *Figure. 1D plot using the default values (w/1000 data point).* … … 2484 2484 The scattering intensity *I(q)* is 2485 2485 2486 .. image:: img/olddocs/image139.PNG2486 .. image:: ..\img\olddocs\image139.PNG 2487 2487 2488 2488 The form factor is 2489 2489 2490 .. image:: img/olddocs/image134.PNG2490 .. image:: ..\img\olddocs\image134.PNG 2491 2491 2492 2492 and the structure factor is 2493 2493 2494 .. image:: img/olddocs/image140.PNG2494 .. image:: ..\img\olddocs\image140.PNG 2495 2495 2496 2496 where 2497 2497 2498 .. image:: img/olddocs/image141.PNG2498 .. image:: ..\img\olddocs\image141.PNG 2499 2499 2500 2500 Here *d* = (repeat) spacing, |delta| = bilayer thickness, the contrast |drho| = SLD(headgroup) - SLD(solvent), … … 2507 2507 The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 2508 2508 2509 .. image:: img/olddocs/image040.gif2509 .. image:: ..\img\olddocs\image040.gif 2510 2510 2511 2511 The returned value is in units of |cm^-1|, on absolute scale. … … 2523 2523 ============== ======== ============= 2524 2524 2525 .. image:: img/olddocs/image142.jpg2525 .. image:: ..\img\olddocs\image142.jpg 2526 2526 2527 2527 *Figure. 1D plot using the default values (w/6000 data point).* … … 2550 2550 The scattering intensity *I(q)* is 2551 2551 2552 .. image:: img/olddocs/image139.PNG2552 .. image:: ..\img\olddocs\image139.PNG 2553 2553 2554 2554 The form factor is 2555 2555 2556 .. image:: img/olddocs/image143.PNG2556 .. image:: ..\img\olddocs\image143.PNG 2557 2557 2558 2558 The structure factor is 2559 2559 2560 .. image:: img/olddocs/image140.PNG2560 .. image:: ..\img\olddocs\image140.PNG 2561 2561 2562 2562 where 2563 2563 2564 .. image:: img/olddocs/image141.PNG2564 .. image:: ..\img\olddocs\image141.PNG 2565 2565 2566 2566 where |delta|\ T = tail length (or *t_length*), |delta|\ H = head thickness (or *h_thickness*), … … 2575 2575 The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 2576 2576 2577 .. image:: img/olddocs/image040.gif2577 .. image:: ..\img\olddocs\image040.gif 2578 2578 2579 2579 The returned value is in units of |cm^-1|, on absolute scale. In the parameters, *sld_tail* = SLD of the tail group, … … 2595 2595 ============== ======== ============= 2596 2596 2597 .. image:: img/olddocs/image144.jpg2597 .. image:: ..\img\olddocs\image144.jpg 2598 2598 2599 2599 *Figure. 1D plot using the default values (w/6000 data point).* … … 2622 2622 The scattering intensity *I(q)* is calculated as 2623 2623 2624 .. image:: img/olddocs/image145.jpg2624 .. image:: ..\img\olddocs\image145.jpg 2625 2625 2626 2626 The form factor of the bilayer is approximated as the cross section of an infinite, planar bilayer of thickness *t* 2627 2627 2628 .. image:: img/olddocs/image146.jpg2628 .. image:: ..\img\olddocs\image146.jpg 2629 2629 2630 2630 Here, the scale factor is used instead of the mass per area of the bilayer (*G*). The scale factor is the volume … … 2635 2635 Non-integer numbers of stacks are calculated as a linear combination of the lower and higher values 2636 2636 2637 .. image:: img/olddocs/image147.jpg2637 .. image:: ..\img\olddocs\image147.jpg 2638 2638 2639 2639 The 2D scattering intensity is the same as 1D, regardless of the orientation of the *q* vector which is defined as 2640 2640 2641 .. image:: img/olddocs/image040.gif2641 .. image:: ..\img\olddocs\image040.gif 2642 2642 2643 2643 The parameters of the model are *Nlayers* = no. of layers, and *pd_spacing* = polydispersity of spacing. … … 2656 2656 ============== ======== ============= 2657 2657 2658 .. image:: img/olddocs/image148.jpg2658 .. image:: ..\img\olddocs\image148.jpg 2659 2659 2660 2660 *Figure. 1D plot using the default values above (w/20000 data point).* … … 2683 2683 The scattering intensity *I(q)* is calculated as 2684 2684 2685 .. image:: img/olddocs/image149.jpg2685 .. image:: ..\img\olddocs\image149.jpg 2686 2686 2687 2687 where *scale* is the volume fraction of spheres, *Vp* is the volume of the primary particle, *V(lattice)* is a volume … … 2695 2695 and nearest neighbor separation *D* is 2696 2696 2697 .. image:: img/olddocs/image150.jpg2697 .. image:: ..\img\olddocs\image150.jpg 2698 2698 2699 2699 The distortion factor (one standard deviation) of the paracrystal is included in the calculation of *Z(q)* 2700 2700 2701 .. image:: img/olddocs/image151.jpg2701 .. image:: ..\img\olddocs\image151.jpg 2702 2702 2703 2703 where *g* is a fractional distortion based on the nearest neighbor distance. … … 2705 2705 The simple cubic lattice is 2706 2706 2707 .. image:: img/olddocs/image152.jpg2707 .. image:: ..\img\olddocs\image152.jpg 2708 2708 2709 2709 For a crystal, diffraction peaks appear at reduced *q*\ -values given by 2710 2710 2711 .. image:: img/olddocs/image153.jpg2711 .. image:: ..\img\olddocs\image153.jpg 2712 2712 2713 2713 where for a simple cubic lattice any *h*\ , *k*\ , *l* are allowed and none are forbidden. Thus the peak positions 2714 2714 correspond to (just the first 5) 2715 2715 2716 .. image:: img/olddocs/image154.jpg2716 .. image:: ..\img\olddocs\image154.jpg 2717 2717 2718 2718 **NB: The calculation of** *Z(q)* **is a double numerical integral that must be carried out with a high density of** … … 2736 2736 default values. 2737 2737 2738 .. image:: img/olddocs/image155.jpg2738 .. image:: ..\img\olddocs\image155.jpg 2739 2739 2740 2740 *Figure. 1D plot in the linear scale using the default values (w/200 data point).* … … 2744 2744 computation. 2745 2745 2746 .. image:: img/olddocs/image156.jpg2747 2748 .. image:: img/olddocs/image157.jpg2746 .. image:: ..\img\olddocs\image156.jpg 2747 2748 .. image:: ..\img\olddocs\image157.jpg 2749 2749 2750 2750 *Figure. 2D plot using the default values (w/200X200 pixels).* … … 2774 2774 The scattering intensity *I(q)* is calculated as 2775 2775 2776 .. image:: img/olddocs/image158.jpg2776 .. image:: ..\img\olddocs\image158.jpg 2777 2777 2778 2778 where *scale* is the volume fraction of spheres, *Vp* is the volume of the primary particle, *V(lattice)* is a volume … … 2786 2786 *R* and nearest neighbor separation *D* is 2787 2787 2788 .. image:: img/olddocs/image159.jpg2788 .. image:: ..\img\olddocs\image159.jpg 2789 2789 2790 2790 The distortion factor (one standard deviation) of the paracrystal is included in the calculation of *Z(q)* 2791 2791 2792 .. image:: img/olddocs/image160.jpg2792 .. image:: ..\img\olddocs\image160.jpg 2793 2793 2794 2794 where *g* is a fractional distortion based on the nearest neighbor distance. … … 2796 2796 The face-centered cubic lattice is 2797 2797 2798 .. image:: img/olddocs/image161.jpg2798 .. image:: ..\img\olddocs\image161.jpg 2799 2799 2800 2800 For a crystal, diffraction peaks appear at reduced q-values given by 2801 2801 2802 .. image:: img/olddocs/image162.jpg2802 .. image:: ..\img\olddocs\image162.jpg 2803 2803 2804 2804 where for a face-centered cubic lattice *h*\ , *k*\ , *l* all odd or all even are allowed and reflections where 2805 2805 *h*\ , *k*\ , *l* are mixed odd/even are forbidden. Thus the peak positions correspond to (just the first 5) 2806 2806 2807 .. image:: img/olddocs/image163.jpg2807 .. image:: ..\img\olddocs\image163.jpg 2808 2808 2809 2809 **NB: The calculation of** *Z(q)* **is a double numerical integral that must be carried out with a high density of** … … 2827 2827 default values. 2828 2828 2829 .. image:: img/olddocs/image164.jpg2829 .. image:: ..\img\olddocs\image164.jpg 2830 2830 2831 2831 *Figure. 1D plot in the linear scale using the default values (w/200 data point).* … … 2835 2835 computation. 2836 2836 2837 .. image:: img/olddocs/image165.gif2838 2839 .. image:: img/olddocs/image166.jpg2837 .. image:: ..\img\olddocs\image165.gif 2838 2839 .. image:: ..\img\olddocs\image166.jpg 2840 2840 2841 2841 *Figure. 2D plot using the default values (w/200X200 pixels).* … … 2865 2865 The scattering intensity *I(q)* is calculated as 2866 2866 2867 .. image:: img/olddocs/image167.jpg2867 .. image:: ..\img\olddocs\image167.jpg 2868 2868 2869 2869 where *scale* is the volume fraction of spheres, *Vp* is the volume of the primary particle, *V(lattice)* is a volume … … 2877 2877 *R* and nearest neighbor separation *D* is 2878 2878 2879 .. image:: img/olddocs/image159.jpg2879 .. image:: ..\img\olddocs\image159.jpg 2880 2880 2881 2881 The distortion factor (one standard deviation) of the paracrystal is included in the calculation of *Z(q)* 2882 2882 2883 .. image:: img/olddocs/image160.jpg2883 .. image:: ..\img\olddocs\image160.jpg 2884 2884 2885 2885 where *g* is a fractional distortion based on the nearest neighbor distance. … … 2887 2887 The body-centered cubic lattice is 2888 2888 2889 .. image:: img/olddocs/image168.jpg2889 .. image:: ..\img\olddocs\image168.jpg 2890 2890 2891 2891 For a crystal, diffraction peaks appear at reduced q-values given by 2892 2892 2893 .. image:: img/olddocs/image162.jpg2893 .. image:: ..\img\olddocs\image162.jpg 2894 2894 2895 2895 where for a body-centered cubic lattice, only reflections where (\ *h* + *k* + *l*\ ) = even are allowed and 2896 2896 reflections where (\ *h* + *k* + *l*\ ) = odd are forbidden. Thus the peak positions correspond to (just the first 5) 2897 2897 2898 .. image:: img/olddocs/image169.jpg2898 .. image:: ..\img\olddocs\image169.jpg 2899 2899 2900 2900 **NB: The calculation of** *Z(q)* **is a double numerical integral that must be carried out with a high density of** … … 2918 2918 default values. 2919 2919 2920 .. image:: img/olddocs/image170.jpg2920 .. image:: ..\img\olddocs\image170.jpg 2921 2921 2922 2922 *Figure. 1D plot in the linear scale using the default values (w/200 data point).* … … 2926 2926 computation. 2927 2927 2928 .. image:: img/olddocs/image165.gif2929 2930 .. image:: img/olddocs/image171.jpg2928 .. image:: ..\img\olddocs\image165.gif 2929 2930 .. image:: ..\img\olddocs\image171.jpg 2931 2931 2932 2932 *Figure. 2D plot using the default values (w/200X200 pixels).* … … 2955 2955 For information about polarised and magnetic scattering, click here_. 2956 2956 2957 .. image:: img/olddocs/image087.jpg2957 .. image:: ..\img\olddocs\image087.jpg 2958 2958 2959 2959 *2.1.37.1. Definition* … … 2962 2962 *b* = *B* / *B* = 1, and *c* = *C* / *B* > 1, the form factor is 2963 2963 2964 .. image:: img/olddocs/image088.PNG2964 .. image:: ..\img\olddocs\image088.PNG 2965 2965 2966 2966 and the contrast is defined as 2967 2967 2968 .. image:: img/olddocs/image089.PNG2968 .. image:: ..\img\olddocs\image089.PNG 2969 2969 2970 2970 The scattering intensity per unit volume is returned in units of |cm^-1|; ie, *I(q)* = |phi| *P(q)*\ . … … 2979 2979 parallel to the *x*-axis of the detector. 2980 2980 2981 .. image:: img/olddocs/image090.jpg2981 .. image:: ..\img\olddocs\image090.jpg 2982 2982 2983 2983 *Figure. Definition of angles for 2D*. 2984 2984 2985 .. image:: img/olddocs/image091.jpg2985 .. image:: ..\img\olddocs\image091.jpg 2986 2986 2987 2987 *Figure. Examples of the angles for oriented pp against the detector plane.* … … 2998 2998 ============== ======== ============= 2999 2999 3000 .. image:: img/olddocs/image092.jpg3000 .. image:: ..\img\olddocs\image092.jpg 3001 3001 3002 3002 *Figure. 1D plot using the default values (w/1000 data point).* … … 3009 3009 angles of |theta|, |phi|, and |psi| respectively). 3010 3010 3011 .. image:: img/olddocs/image093.gif3011 .. image:: ..\img\olddocs\image093.gif 3012 3012 3013 3013 *Figure. Comparison between 1D and averaged 2D.* … … 3043 3043 dimensions *A*, *B*, *C* such that *A* < *B* < *C*. 3044 3044 3045 .. image:: img/olddocs/image087.jpg3045 .. image:: ..\img\olddocs\image087.jpg 3046 3046 3047 3047 There are rectangular "slabs" of thickness *tA* that add to the *A* dimension (on the *BC* faces). There are similar 3048 3048 slabs on the *AC* (= *tB*) and *AB* (= *tC*) faces. The projection in the *AB* plane is then 3049 3049 3050 .. image:: img/olddocs/image094.jpg3050 .. image:: ..\img\olddocs\image094.jpg 3051 3051 3052 3052 The volume of the solid is 3053 3053 3054 .. image:: img/olddocs/image095.PNG3054 .. image:: ..\img\olddocs\image095.PNG 3055 3055 3056 3056 **meaning that there are "gaps" at the corners of the solid.** … … 3084 3084 parallel to the *x*-axis of the detector. 3085 3085 3086 .. image:: img/olddocs/image090.jpg3086 .. image:: ..\img\olddocs\image090.jpg 3087 3087 3088 3088 *Figure. Definition of angles for 2D*. 3089 3089 3090 .. image:: img/olddocs/image091.jpg3090 .. image:: ..\img\olddocs\image091.jpg 3091 3091 3092 3092 *Figure. Examples of the angles for oriented cspp against the detector plane.* … … 3113 3113 ============== ======== ============= 3114 3114 3115 .. image:: img/olddocs/image096.jpg3115 .. image:: ..\img\olddocs\image096.jpg 3116 3116 3117 3117 *Figure. 1D plot using the default values (w/256 data points).* 3118 3118 3119 .. image:: img/olddocs/image097.jpg3119 .. image:: ..\img\olddocs\image097.jpg 3120 3120 3121 3121 *Figure. 2D plot using the default values (w/(256X265) data points).* … … 3398 3398 calculation. **NB: No size polydispersity is included in this model, use the** Poly_GaussCoil_ **Model instead** 3399 3399 3400 .. image:: img/olddocs/image172.PNG3400 .. image:: ..\img\olddocs\image172.PNG 3401 3401 3402 3402 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 3403 3403 3404 .. image:: img/olddocs/image040.gif3404 .. image:: ..\img\olddocs\image040.gif 3405 3405 3406 3406 ============== ======== ============= … … 3412 3412 ============== ======== ============= 3413 3413 3414 .. image:: img/olddocs/image173.jpg3414 .. image:: ..\img\olddocs\image173.jpg 3415 3415 3416 3416 *Figure. 1D plot using the default values (w/200 data point).* … … 3439 3439 The scattering intensity *I(q)* is calculated as 3440 3440 3441 .. image:: img/olddocs/image174.jpg3441 .. image:: ..\img\olddocs\image174.jpg 3442 3442 3443 3443 Here the peak position is related to the d-spacing as *Q0* = 2|pi| / *d0*. … … 3445 3445 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 3446 3446 3447 .. image:: img/olddocs/image040.gif3447 .. image:: ..\img\olddocs\image040.gif 3448 3448 3449 3449 ================== ======== ============= … … 3459 3459 ================== ======== ============= 3460 3460 3461 .. image:: img/olddocs/image175.jpg3461 .. image:: ..\img\olddocs\image175.jpg 3462 3462 3463 3463 *Figure. 1D plot using the default values (w/200 data point).* … … 3483 3483 The scattering intensity *I(q)* is calculated as 3484 3484 3485 .. image:: img/olddocs/image176.jpg3485 .. image:: ..\img\olddocs\image176.jpg 3486 3486 3487 3487 The first term describes Porod scattering from clusters (exponent = n) and the second term is a Lorentzian function … … 3494 3494 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 3495 3495 3496 .. image:: img/olddocs/image040.gif3496 .. image:: ..\img\olddocs\image040.gif 3497 3497 3498 3498 ==================== ======== ============= … … 3507 3507 ==================== ======== ============= 3508 3508 3509 .. image:: img/olddocs/image177.jpg3509 .. image:: ..\img\olddocs\image177.jpg 3510 3510 3511 3511 *Figure. 1D plot using the default values (w/500 data points).* … … 3528 3528 The Ornstein-Zernicke model is defined by 3529 3529 3530 .. image:: img/olddocs/image178.PNG3530 .. image:: ..\img\olddocs\image178.PNG 3531 3531 3532 3532 The parameter *L* is the screening length. … … 3534 3534 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 3535 3535 3536 .. image:: img/olddocs/image040.gif3536 .. image:: ..\img\olddocs\image040.gif 3537 3537 3538 3538 ============== ======== ============= … … 3544 3544 ============== ======== ============= 3545 3545 3546 .. image:: img/olddocs/image179.jpg3546 .. image:: ..\img\olddocs\image179.jpg 3547 3547 3548 3548 * Figure. 1D plot using the default values (w/200 data point).* … … 3567 3567 *2.2.5.1. Definition* 3568 3568 3569 .. image:: img/olddocs/image180_corrected.PNG3569 .. image:: ..\img\olddocs\image180_corrected.PNG 3570 3570 3571 3571 The parameter *L* is the correlation length. … … 3573 3573 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 3574 3574 3575 .. image:: img/olddocs/image040.gif3575 .. image:: ..\img\olddocs\image040.gif 3576 3576 3577 3577 ============== ======== ============= … … 3583 3583 ============== ======== ============= 3584 3584 3585 .. image:: img/olddocs/image181.jpg3585 .. image:: ..\img\olddocs\image181.jpg 3586 3586 3587 3587 * Figure. 1D plot using the default values (w/200 data point).* … … 3604 3604 This model describes a simple power law with background. 3605 3605 3606 .. image:: img/olddocs/image182.PNG3606 .. image:: ..\img\olddocs\image182.PNG 3607 3607 3608 3608 Note the minus sign in front of the exponent. The parameter *m* should therefore be entered as a **positive** number. … … 3616 3616 ============== ======== ============= 3617 3617 3618 .. image:: img/olddocs/image183.jpg3618 .. image:: ..\img\olddocs\image183.jpg 3619 3619 3620 3620 *Figure. 1D plot using the default values (w/200 data point).* … … 3635 3635 *2.2.7.1. Definition* 3636 3636 3637 .. image:: img/olddocs/image184.PNG3637 .. image:: ..\img\olddocs\image184.PNG 3638 3638 3639 3639 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 3640 3640 3641 .. image:: img/olddocs/image040.gif3641 .. image:: ..\img\olddocs\image040.gif 3642 3642 3643 3643 ============== ======== ============= … … 3650 3650 ============== ======== ============= 3651 3651 3652 .. image:: img/olddocs/image185.jpg3652 .. image:: ..\img\olddocs\image185.jpg 3653 3653 3654 3654 *Figure. 1D plot using the default values (w/200 data point).* … … 3673 3673 *2.2.8.1. Definition* 3674 3674 3675 .. image:: img/olddocs/image186.PNG3675 .. image:: ..\img\olddocs\image186.PNG 3676 3676 3677 3677 The *scale* parameter is the volume fraction of the building blocks, *R0* is the radius of the building block, *Df* is … … 3683 3683 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 3684 3684 3685 .. image:: img/olddocs/image040.gif3685 .. image:: ..\img\olddocs\image040.gif 3686 3686 3687 3687 ============== ======== ============= … … 3697 3697 ============== ======== ============= 3698 3698 3699 .. image:: img/olddocs/image187.jpg3699 .. image:: ..\img\olddocs\image187.jpg 3700 3700 3701 3701 *Figure. 1D plot using the default values (w/200 data point).* … … 3715 3715 *2.2.9.1. Definition* 3716 3716 3717 .. image:: img/olddocs/mass_fractal_eq1.jpg3717 .. image:: ..\img\olddocs\mass_fractal_eq1.jpg 3718 3718 3719 3719 where *R* is the radius of the building block, *Dm* is the **mass** fractal dimension, |zeta| is the cut-off length, … … 3734 3734 ============== ======== ============= 3735 3735 3736 .. image:: img/olddocs/mass_fractal_fig1.jpg3736 .. image:: ..\img\olddocs\mass_fractal_fig1.jpg 3737 3737 3738 3738 *Figure. 1D plot using default values.* … … 3755 3755 *2.2.10.1. Definition* 3756 3756 3757 .. image:: img/olddocs/surface_fractal_eq1.gif3757 .. image:: ..\img\olddocs\surface_fractal_eq1.gif 3758 3758 3759 3759 where *R* is the radius of the building block, *Ds* is the **surface** fractal dimension, |zeta| is the cut-off length, … … 3774 3774 ============== ======== ============= 3775 3775 3776 .. image:: img/olddocs/surface_fractal_fig1.jpg3776 .. image:: ..\img\olddocs\surface_fractal_fig1.jpg 3777 3777 3778 3778 *Figure. 1D plot using default values.* … … 3802 3802 The scattered intensity *I(q)* is calculated using a modified Ornstein-Zernicke equation 3803 3803 3804 .. image:: img/olddocs/masssurface_fractal_eq1.jpg3804 .. image:: ..\img\olddocs\masssurface_fractal_eq1.jpg 3805 3805 3806 3806 where *Rg* is the size of the cluster, *rg* is the size of the primary particle, *Ds* is the surface fractal dimension, … … 3822 3822 ============== ======== ============= 3823 3823 3824 .. image:: img/olddocs/masssurface_fractal_fig1.jpg3824 .. image:: ..\img\olddocs\masssurface_fractal_fig1.jpg 3825 3825 3826 3826 *Figure. 1D plot using default values.* … … 3848 3848 *2.2.12.1. Definition* 3849 3849 3850 .. image:: img/olddocs/fractcore_eq1.gif3850 .. image:: ..\img\olddocs\fractcore_eq1.gif 3851 3851 3852 3852 The form factor *P(q)* is that from CoreShellModel_ with *bkg* = 0 3853 3853 3854 .. image:: img/olddocs/image013.PNG3854 .. image:: ..\img\olddocs\image013.PNG 3855 3855 3856 3856 while the fractal structure factor S(q) is 3857 3857 3858 .. image:: img/olddocs/fractcore_eq3.gif3858 .. image:: ..\img\olddocs\fractcore_eq3.gif 3859 3859 3860 3860 where *Df* = frac_dim, |xi| = cor_length, *rc* = (core) radius, and *scale* = volume fraction. … … 3864 3864 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 3865 3865 3866 .. image:: img/olddocs/image040.gif3866 .. image:: ..\img\olddocs\image040.gif 3867 3867 3868 3868 ============== ======== ============= … … 3880 3880 ============== ======== ============= 3881 3881 3882 .. image:: img/olddocs/image188.jpg3882 .. image:: ..\img\olddocs\image188.jpg 3883 3883 3884 3884 *Figure. 1D plot using the default values (w/500 data points).* … … 3905 3905 The scattering intensity *I(q)* is calculated as (eqn 5 from the reference) 3906 3906 3907 .. image:: img/olddocs/image189.jpg3907 .. image:: ..\img\olddocs\image189.jpg 3908 3908 3909 3909 |bigzeta| is the length scale of the static correlations in the gel, which can be attributed to the "frozen-in" … … 3917 3917 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 3918 3918 3919 .. image:: img/olddocs/image040.gif3919 .. image:: ..\img\olddocs\image040.gif 3920 3920 3921 3921 =================================== ======== ============= … … 3929 3929 =================================== ======== ============= 3930 3930 3931 .. image:: img/olddocs/image190.jpg3931 .. image:: ..\img\olddocs\image190.jpg 3932 3932 3933 3933 *Figure. 1D plot using the default values (w/500 data points).* … … 3949 3949 *2.2.14.1. Definition* 3950 3950 3951 .. image:: img/olddocs/image191.PNG3951 .. image:: ..\img\olddocs\image191.PNG 3952 3952 3953 3953 where *K* is the contrast factor for the polymer, *Lb* is the Bjerrum length, *h* is the virial parameter, *b* is the … … 3957 3957 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 3958 3958 3959 .. image:: img/olddocs/image040.gif3959 .. image:: ..\img\olddocs\image040.gif 3960 3960 3961 3961 ============== ======== ============= … … 3992 3992 This model fits the Guinier function 3993 3993 3994 .. image:: img/olddocs/image192.PNG3994 .. image:: ..\img\olddocs\image192.PNG 3995 3995 3996 3996 to the data directly without any need for linearisation (*cf*. Ln *I(q)* vs *q*\ :sup:`2`). … … 3998 3998 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 3999 3999 4000 .. image:: img/olddocs/image040.gif4000 .. image:: ..\img\olddocs\image040.gif 4001 4001 4002 4002 ============== ======== ============= … … 4027 4027 The following functional form is used 4028 4028 4029 .. image:: img/olddocs/image193.jpg4029 .. image:: ..\img\olddocs\image193.jpg 4030 4030 4031 4031 This is based on the generalized Guinier law for such elongated objects (see the Glatter reference below). For 3D … … 4036 4036 Enforcing the continuity of the Guinier and Porod functions and their derivatives yields 4037 4037 4038 .. image:: img/olddocs/image194.jpg4038 .. image:: ..\img\olddocs\image194.jpg 4039 4039 4040 4040 and 4041 4041 4042 .. image:: img/olddocs/image195.jpg4042 .. image:: ..\img\olddocs\image195.jpg 4043 4043 4044 4044 Note that … … 4052 4052 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 4053 4053 4054 .. image:: img/olddocs/image008.PNG4054 .. image:: ..\img\olddocs\image008.PNG 4055 4055 4056 4056 ============================== ======== ============= … … 4064 4064 ============================== ======== ============= 4065 4065 4066 .. image:: img/olddocs/image196.jpg4066 .. image:: ..\img\olddocs\image196.jpg 4067 4067 4068 4068 *Figure. 1D plot using the default values (w/500 data points).* … … 4083 4083 This model fits the Porod function 4084 4084 4085 .. image:: img/olddocs/image197_corrected.PNG4085 .. image:: ..\img\olddocs\image197_corrected.PNG 4086 4086 4087 4087 to the data directly without any need for linearisation (*cf*. Log *I(q)* vs Log *q*). … … 4092 4092 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 4093 4093 4094 .. image:: img/olddocs/image040.gif4094 .. image:: ..\img\olddocs\image040.gif 4095 4095 4096 4096 ============== ======== ============= … … 4113 4113 This model describes a Gaussian shaped peak on a flat background 4114 4114 4115 .. image:: img/olddocs/image198.PNG4115 .. image:: ..\img\olddocs\image198.PNG 4116 4116 4117 4117 with the peak having height of *I0* centered at *q0* and having a standard deviation of *B*. The FWHM (full-width … … 4120 4120 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 4121 4121 4122 .. image:: img/olddocs/image040.gif4122 .. image:: ..\img\olddocs\image040.gif 4123 4123 4124 4124 ============== ======== ============= … … 4131 4131 ============== ======== ============= 4132 4132 4133 .. image:: img/olddocs/image199.jpg4133 .. image:: ..\img\olddocs\image199.jpg 4134 4134 4135 4135 *Figure. 1D plot using the default values (w/500 data points).* … … 4147 4147 This model describes a Lorentzian shaped peak on a flat background 4148 4148 4149 .. image:: img/olddocs/image200.PNG4149 .. image:: ..\img\olddocs\image200.PNG 4150 4150 4151 4151 with the peak having height of *I0* centered at *q0* and having a HWHM (half-width half-maximum) of B. … … 4153 4153 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 4154 4154 4155 .. image:: img/olddocs/image040.gif4155 .. image:: ..\img\olddocs\image040.gif 4156 4156 4157 4157 ============== ======== ============= … … 4164 4164 ============== ======== ============= 4165 4165 4166 .. image:: img/olddocs/image201.jpg4166 .. image:: ..\img\olddocs\image201.jpg 4167 4167 4168 4168 *Figure. 1D plot using the default values (w/500 data points).* … … 4188 4188 The scattering intensity *I(q)* is calculated as 4189 4189 4190 .. image:: img/olddocs/image202.PNG4190 .. image:: ..\img\olddocs\image202.PNG 4191 4191 4192 4192 where the dimensionless chain dimension is 4193 4193 4194 .. image:: img/olddocs/image203.PNG4194 .. image:: ..\img\olddocs\image203.PNG 4195 4195 4196 4196 and the polydispersity is 4197 4197 4198 .. image:: img/olddocs/image204.PNG4198 .. image:: ..\img\olddocs\image204.PNG 4199 4199 4200 4200 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 4201 4201 4202 .. image:: img/olddocs/image040.gif4202 .. image:: ..\img\olddocs\image040.gif 4203 4203 4204 4204 This example dataset is produced using 200 data points, using 200 data points, … … 4214 4214 ============== ======== ============= 4215 4215 4216 .. image:: img/olddocs/image205.jpg4216 .. image:: ..\img\olddocs\image205.jpg 4217 4217 4218 4218 *Figure. 1D plot using the default values (w/200 data point).* … … 4240 4240 The form factor was originally presented in the following integral form (Benoit, 1957) 4241 4241 4242 .. image:: img/olddocs/image206.jpg4242 .. image:: ..\img\olddocs\image206.jpg 4243 4243 4244 4244 where |nu| is the excluded volume parameter (which is related to the Porod exponent *m* as |nu| = 1 / *m*), *a* is the … … 4246 4246 into an almost analytical form as follows (Hammouda, 1993) 4247 4247 4248 .. image:: img/olddocs/image207.jpg4248 .. image:: ..\img\olddocs\image207.jpg 4249 4249 4250 4250 where |gamma|\ *(x,U)* is the incomplete gamma function 4251 4251 4252 .. image:: img/olddocs/image208.jpg4252 .. image:: ..\img\olddocs\image208.jpg 4253 4253 4254 4254 and the variable *U* is given in terms of the scattering vector *Q* as 4255 4255 4256 .. image:: img/olddocs/image209.jpg4256 .. image:: ..\img\olddocs\image209.jpg 4257 4257 4258 4258 The square of the radius-of-gyration is defined as 4259 4259 4260 .. image:: img/olddocs/image210.jpg4260 .. image:: ..\img\olddocs\image210.jpg 4261 4261 4262 4262 Note that this model applies only in the mass fractal range (ie, 5/3 <= *m* <= 3) and **does not** apply to surface … … 4266 4266 A low-*Q* expansion yields the Guinier form and a high-*Q* expansion yields the Porod form which is given by 4267 4267 4268 .. image:: img/olddocs/image211.jpg4268 .. image:: ..\img\olddocs\image211.jpg 4269 4269 4270 4270 Here |biggamma|\ *(x)* = |gamma|\ *(x,inf)* is the gamma function. … … 4272 4272 The asymptotic limit is dominated by the first term 4273 4273 4274 .. image:: img/olddocs/image212.jpg4274 .. image:: ..\img\olddocs\image212.jpg 4275 4275 4276 4276 The special case when |nu| = 0.5 (or *m* = 1/|nu| = 2) corresponds to Gaussian chains for which the form factor is given 4277 4277 by the familiar Debye_ function. 4278 4278 4279 .. image:: img/olddocs/image213.jpg4279 .. image:: ..\img\olddocs\image213.jpg 4280 4280 4281 4281 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 4282 4282 4283 .. image:: img/olddocs/image040.gif4283 .. image:: ..\img\olddocs\image040.gif 4284 4284 4285 4285 This example dataset is produced using 200 data points, *qmin* = 0.001 |Ang^-1|, *qmax* = 0.2 |Ang^-1| and the default … … 4295 4295 =================== ======== ============= 4296 4296 4297 .. image:: img/olddocs/image214.jpg4297 .. image:: ..\img\olddocs\image214.jpg 4298 4298 4299 4299 *Figure. 1D plot using the default values (w/500 data points).* … … 4379 4379 ======================= ======== ============= 4380 4380 4381 .. image:: img/olddocs/image215.jpg4381 .. image:: ..\img\olddocs\image215.jpg 4382 4382 4383 4383 *Figure. 1D plot using the default values (w/500 data points).* … … 4401 4401 The scattering intensity *I(q)* is calculated as 4402 4402 4403 .. image:: img/olddocs/image216.jpgÂ4403 .. image:: ..\img\olddocs\image216.jpg 4404 4404 4405 4405 where *A* = Lorentzian scale factor #1, *C* = Lorentzian scale #2, |xi|\ :sub:`1` and |xi|\ :sub:`2` are the … … 4409 4409 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 4410 4410 4411 .. image:: img/olddocs/image040.gif4411 .. image:: ..\img\olddocs\image040.gif 4412 4412 4413 4413 =============================== ======== ============= … … 4423 4423 =============================== ======== ============= 4424 4424 4425 .. image:: img/olddocs/image217.jpg4425 .. image:: ..\img\olddocs\image217.jpg 4426 4426 4427 4427 *Figure. 1D plot using the default values (w/500 data points).* … … 4445 4445 The scattering intensity *I(q)* is calculated as 4446 4446 4447 .. image:: img/olddocs/image218.jpg4447 .. image:: ..\img\olddocs\image218.jpg 4448 4448 4449 4449 where *qc* is the location of the crossover from one slope to the other. The scaling *coef_A* sets the overall … … 4455 4455 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 4456 4456 4457 .. image:: img/olddocs/image040.gif4457 .. image:: ..\img\olddocs\image040.gif 4458 4458 4459 4459 ============== ======== ============= … … 4467 4467 ============== ======== ============= 4468 4468 4469 .. image:: img/olddocs/image219.jpg4469 .. image:: ..\img\olddocs\image219.jpg 4470 4470 4471 4471 *Figure. 1D plot using the default values (w/500 data points).* … … 4496 4496 The empirical fit function is 4497 4497 4498 .. image:: img/olddocs/image220.jpg4498 .. image:: ..\img\olddocs\image220.jpg 4499 4499 4500 4500 For each level, the four parameters *Gi*, *Rg,i*, *Bi* and *Pi* must be chosen. … … 4507 4507 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 4508 4508 4509 .. image:: img/olddocs/image040.gif4509 .. image:: ..\img\olddocs\image040.gif 4510 4510 4511 4511 ============== ======== ============= … … 4524 4524 ============== ======== ============= 4525 4525 4526 .. image:: img/olddocs/image221.jpg4526 .. image:: ..\img\olddocs\image221.jpg 4527 4527 4528 4528 *Figure. 1D plot using the default values (w/500 data points).* … … 4542 4542 This calculates the simple linear function 4543 4543 4544 .. image:: img/olddocs/image222.PNG4544 .. image:: ..\img\olddocs\image222.PNG 4545 4545 4546 4546 **NB: For 2D plots,** *I(q)* = *I(qx)*\ *\ *I(qy)*, **which is a different definition to other shape independent models.** … … 4577 4577 The scattered intensity *I(q)* is calculated as 4578 4578 4579 .. image:: img/olddocs/image233.gif4579 .. image:: ..\img\olddocs\image233.gif 4580 4580 4581 4581 where 4582 4582 4583 .. image:: img/olddocs/image234.gif4583 .. image:: ..\img\olddocs\image234.gif 4584 4584 4585 4585 Note that the first term reduces to the Ornstein-Zernicke equation when *D* = 2; ie, when the Flory exponent is 0.5 … … 4597 4597 ============================ ======== ============= 4598 4598 4599 .. image:: img/olddocs/image235.gif4599 .. image:: ..\img\olddocs\image235.gif 4600 4600 4601 4601 *Figure. 1D plot using the default values (w/300 data points).* … … 4619 4619 For a star with *f* arms: 4620 4620 4621 .. image:: img/olddocs/star1.png4621 .. image:: ..\img\olddocs\star1.png 4622 4622 4623 4623 where 4624 4624 4625 .. image:: img/olddocs/star2.png4625 .. image:: ..\img\olddocs\star2.png 4626 4626 4627 4627 and 4628 4628 4629 .. image:: img/olddocs/star3.png4629 .. image:: ..\img\olddocs\star3.png 4630 4630 4631 4631 is the square of the ensemble average radius-of-gyration of an arm. … … 4656 4656 Also see ReflectivityIIModel_. 4657 4657 4658 .. image:: img/olddocs/image231.bmp4658 .. image:: ..\img\olddocs\image231.bmp 4659 4659 4660 4660 *Figure. Comparison (using the SLD profile below) with the NIST web calculation (circles)* 4661 4661 http://www.ncnr.nist.gov/resources/reflcalc.html 4662 4662 4663 .. image:: img/olddocs/image232.gif4663 .. image:: ..\img\olddocs\image232.gif 4664 4664 4665 4665 *Figure. SLD profile used for the calculation (above).* … … 4685 4685 1) Erf 4686 4686 4687 .. image:: img/olddocs/image051.gif4687 .. image:: ..\img\olddocs\image051.gif 4688 4688 4689 4689 2) Power-Law 4690 4690 4691 .. image:: img/olddocs/image050.gif4691 .. image:: ..\img\olddocs\image050.gif 4692 4692 4693 4693 3) Exp 4694 4694 4695 .. image:: img/olddocs/image049.gif4695 .. image:: ..\img\olddocs\image049.gif 4696 4696 4697 4697 The constant *A* in the expressions above (but the parameter *nu* in the model!) is an input. … … 4717 4717 The calculation uses the Percus-Yevick closure where the interparticle potential is 4718 4718 4719 .. image:: img/olddocs/image223.PNG4719 .. image:: ..\img\olddocs\image223.PNG 4720 4720 4721 4721 where *r* is the distance from the center of the sphere of a radius *R*. … … 4723 4723 For a 2D plot, the wave transfer is defined as 4724 4724 4725 .. image:: img/olddocs/image040.gif4725 .. image:: ..\img\olddocs\image040.gif 4726 4726 4727 4727 ============== ======== ============= … … 4732 4732 ============== ======== ============= 4733 4733 4734 .. image:: img/olddocs/image224.jpg4734 .. image:: ..\img\olddocs\image224.jpg 4735 4735 4736 4736 *Figure. 1D plot using the default values (in linear scale).* … … 4758 4758 The interaction potential is: 4759 4759 4760 .. image:: img/olddocs/image225.PNG4760 .. image:: ..\img\olddocs\image225.PNG 4761 4761 4762 4762 where *r* is the distance from the center of the sphere of a radius *R*. … … 4764 4764 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 4765 4765 4766 .. image:: img/olddocs/image040.gif4766 .. image:: ..\img\olddocs\image040.gif 4767 4767 4768 4768 ============== ========= ============= … … 4775 4775 ============== ========= ============= 4776 4776 4777 .. image:: img/olddocs/image226.jpg4777 .. image:: ..\img\olddocs\image226.jpg 4778 4778 4779 4779 *Figure. 1D plot using the default values (in linear scale).* … … 4803 4803 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 4804 4804 4805 .. image:: img/olddocs/image040.gif4805 .. image:: ..\img\olddocs\image040.gif 4806 4806 4807 4807 ============== ======== ============= … … 4816 4816 ============== ======== ============= 4817 4817 4818 .. image:: img/olddocs/image227.jpg4818 .. image:: ..\img\olddocs\image227.jpg 4819 4819 4820 4820 *Figure. 1D plot using the default values (in linear scale).* … … 4842 4842 that smaller |tau| means stronger attraction. 4843 4843 4844 .. image:: img/olddocs/image228.PNG4844 .. image:: ..\img\olddocs\image228.PNG 4845 4845 4846 4846 where the interaction potential is 4847 4847 4848 .. image:: img/olddocs/image229.PNG4848 .. image:: ..\img\olddocs\image229.PNG 4849 4849 4850 4850 The Percus-Yevick (PY) closure was used for this calculation, and is an adequate closure for an attractive interparticle … … 4862 4862 For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as 4863 4863 4864 .. image:: img/olddocs/image040.gif4864 .. image:: ..\img\olddocs\image040.gif 4865 4865 4866 4866 ============== ======== ============= … … 4873 4873 ============== ======== ============= 4874 4874 4875 .. image:: img/olddocs/image230.jpg4875 .. image:: ..\img\olddocs\image230.jpg 4876 4876 4877 4877 *Figure. 1D plot using the default values (in linear scale).*
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